Following [1]}, walls inside \(\Theta \)  are of various types denoted 0-III depending on the birational geometry of the moduli spaces near the wall. Walls of Type I correspond to flops induced by curves. By [2]}, every flop in a single exceptional \((-1,-1)\) -curve can be realised by a wall-crossing of Type I directly from \(\mathfrak {C}_0\) , which is very much not true for other resolutions; see [2]}. Walls of Type III – that contract a divisor to a curve – arise from exceptional \((-2,0)\) -curves corresponding to certain “boundary” edges in the triangulation for \(G\) -Hilb. We will define this terminology in Section REF . The final possibility is that an exceptional curve is a \((1,-3)\) -curve that, if the corresponding inequality was irredundant, would produce a wall of Type II where by definition a divisor is contracted to a point. [2]} shows that there are in fact no Type II walls in \(\Theta \) .
